Semirigidity and the enumeration of nilpotent semigroups of index three

TL;DR

The paper develops a comprehensive framework for counting 3-nilpotent semigroups of a fixed order by encoding semigroups as N(X,K,m) and translating the problem into combinatorics of partial partitions and orbit counting under S_r. It introduces semirigidity as a computationally tractable relaxation of rigidity, derives explicit formulas and bounds for counts up to identity, up to isomorphism, and up to equivalence (including commutative and self-dual cases), and uses Burnside's lemma and frieze constructions to bound orbit numbers. The approach yields practical, provable bounds that match known exact counts for small n and align with results in prior literature (KRS, DM, JMS), with extensive GAP computations up to n=10. Overall, the work provides both theoretical insight and computable methods for enumerating nilpotent semigroups, highlighting the dominance of rigid structures in the asymptotics and offering a scalable path to more refined classifications (commutative, dual, and equivalence classes).

Abstract

There is strong evidence for the belief that `almost all' finite semigroups, whether we consider multiplication operations on a fixed set or their isomorphism classes, are nilpotent of index 3 (3-nilpotent for short). The only known method for counting all semigroups of given order is exhaustive testing, but formulae exist for the numbers of 3-nilpotent ones, and it is also known that `almost all' of these are rigid (have only trivial automorphism). Here we express the number of distinct 3-nilpotent semigroup operations on a fixed set of cardinality $n$ as a sum of Stirling numbers, and provide a new expression for the number of isomorphism classes of 3-nilpotent semigroups of cardinality $n$. We introduce a notion of semirigidity for semigroups (as a generalization of rigidity) and find computationally tractable formulae giving an upper bound for the number of pairwise non-isomorphic semirigid 3-nilpotent semigroups, and thus an improved lower bound for the number of all 3-nilpotent semigroups up to isomorphism. Analogous formulae are also developed for isomorphism classes such as commutative and self-dual semigroups, and for equivalence classes (isomorphic or anti-isomorphic). The method relies on an application of the theory of orbit counting in permutation group actions. Our main results are accompanied by tables containing values of these numbers and bounds up to $n=10$ with computations carried out in GAP (but perfectly feasible well beyond this value of $n$).

Theorems & Definitions (49)

• Corollary 1.2
• proof
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 3.1
• proof
• Lemma 4.1
• proof